Probability theory and stochastic processes Definition

Local time

Builds on Wiener process Probability space

Definition

Let $(B_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, P)$ . The local time of $B$ is a random field $L = (L(t, x))_{t \ge 0, \, x \in \R}$ such that, almost surely:

the map $(t, x) \mapsto L(t, x)$ is continuous;
for every $t \ge 0$ and every bounded Borel function $f \colon \R \to \R$ , $\int_0^t f(B_s) \, ds = \int_{\R} f(x) \, L(t, x) \, dx.$

Such a field exists, and it is unique up to indistinguishability. For each $t$ and $x$ it satisfies, almost surely, $L(t, x) = \lim_{\varepsilon \downarrow 0} \frac{1}{2\varepsilon} \int_0^t \mathbf{1}_{\set{\abs{B_s - x} < \varepsilon}} \, ds.$

Remarks

Example: $L(0, x) = 0$ for all $x$ , and $E[L(t, 0)] = \int_0^t (2\pi s)^{-1/2} \, ds = \sqrt{2t/\pi}$ , because the density of $B_s$ at $0$ is $(2\pi s)^{-1/2}$ . Non-example: the field $\int_0^t \mathbf{1}_{\set{B_s = x}} \, ds$ is almost surely $0$ for every $x$ , so it fails clause 2 with $f = 1$ and $t > 0$ . For fixed $x$ , the map $t \mapsto L(t, x)$ is nondecreasing and increases only on the set of times $t$ with $B_t = x$ . The notations $L_t^x$ and $\ell_t(x)$ are also common. With this normalisation, $L(t, 0)$ has the same law as $\max_{s \le t} B_s$ .

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Probability space concept
Wiener process concept
Local time definition

Tag	`C8W7J0S73V`
URI	`m2://C8W7J0S73V`
Kind	definition
Writing	W0 stub
MSC	60J55, 60J65
Defines	Local time
Uses	Wiener process, Probability space
Changed	2 November 2026

Local time

Definition

Remarks

Reading path

Further reading

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